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Questions tagged [real-analysis]

For questions about real analysis, a branch of mathematics dealing with limits, convergence of sequences, construction of the real numbers, least upper bound property and related analysis topics, such as continuity, differentiation, and integration through the Fundamental Theorem of Calculus.

0
votes
1answer
42 views

Proving this integral as a function of $t$ is continuous?

So let $h$ be a function bounded by $M$, so that $|h|< M$. Also let $g$ be a continuous and non-negative functions. Define: $$f(t) = \int_0^t g(u-t)h(u) du$$ How can I show $f(t)$ is continuous ...
3
votes
1answer
37 views

Show the terms of a sequence $u_n$ is $0$.

Given $u_n=\sum_{k=1}^{\infty} u_{n+k}^2$ and $\sum_{n=1}^{\infty}u_n$ converge. Show $u_k=0 \forall k \in \mathbb{N}$. Remark:This sequence is not increasing as $u_n-u_{n+1}=u_{n+1}^2\geq0$. It is ...
1
vote
1answer
32 views

Uniform convergence of iterated improper integrals on $(0,\infty)$

I'm trying to get a better understanding of when it is permissible to swtich conditionally convergent improper integrals (when Fubini inapplicable) and I looked at a case where it works: $$\int_0^\...
0
votes
1answer
26 views

Proving that a differentiable function f, with f' bounded, is uniformly continuous

Let $f: R \rightarrow R$ be a differentiable function. Prove that if $f'$ is bounded, then $f$ is uniformly continuous My attempt: Since $f$ is differentiable, we have $f'(x) = \lim_{x \to x_0}\...
0
votes
4answers
54 views

Prove that $F$ is closed

We are given that if a sequence $(x_n)\subset F$ converges to some point $x\in M$, then $x\in F$. We must prove that $F$ is closed. My attempt: I strongly believe that the trick here is to come up ...
0
votes
1answer
28 views

If $f: I \rightarrow \mathbb{R}$ is continuously differentiable at $a$, then $\lim_{x\rightarrow a} \frac{|R(a,x)|}{|x-a|} = 0$.

Where $R(a,x) = f(x) - f(a) - f'(a)(x-a)$. Via the "reverse" triangle inequality, wouldn't the limit be $\geq 0$? So $\lim_{x\rightarrow a} \frac{|R(a,x)|}{|x-a|} = \lim_{x\rightarrow a} \frac{|f(x) ...
0
votes
0answers
23 views

Suppose that $f_k$ is a sequence of differentiable functions that converges uniformly to a function $f$. Must $f$ be differentiable? [duplicate]

Suppose that $f_k: (0,1) \rightarrow \mathbb{R}$ is a sequence of differentiable functions that converges uniformly to a function $f: (0,1) \rightarrow \mathbb{R}$. Must $f$ be differentiable? So I'm ...
0
votes
0answers
12 views

some reference requests for Borel algebras

I believe the following are all true, and I could probably prove them myself if necessary. However this would be inelegant, and I would much prefer just to have references. I have been Googling ...
2
votes
3answers
46 views

If two functions are equal almost everywhere, the first is continuous a.e., is the second?

If $f = g$ a.e. in $E \in \mathfrak{M}$ (the Lebesgue measurable sets) and $f$ is continuous a.e. in $E$, is $g$ continuous a.e. in $E$? I think this is true. My “proof”: Let us denote $D_1 = \...
0
votes
1answer
38 views

let $f(x) = \tan ^{-1}x, x \in \mathbb{R} $ .then choose the correct statement

let $f(x) = \tan ^{-1}x, x \in \mathbb{R} $ .then choose the correct statement $1.$ there exist a polynomial $p(x)$ satisfying $p(x)f'(x) =1$ for all $x$ $2.$$f^{(n)}(0) =0$ for all ...
3
votes
1answer
46 views

Showing $f(x) = x^{-1/2}$ is Lebesgue integrable on $(0,1]$

While reviewing for an exam recently, I came across this question which gave me pause. Explain why $f(x) = x^{\frac{-1}{2}}$ is Lebesgue integrable over $(0,1]$. It is clear that $f$ is a ...
0
votes
1answer
35 views

Show that $f(x,y) = \cos(x+y)+8$ is continuous at $(0,0).$

I tried the following for this problem: We first compute $f(0,0) = 9.$ Next we look at $$|f(x,y)- f(0,0)| = |\cos(x+y)-1|\leq \frac{(x+y)^2}{2}$$ when $(x,y)$ is in the neighbourhood of $(0,0).$ ...
2
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0answers
25 views

Open and closed operators.

I was wondering whether there is a general methodology of going about showing whether bounded, linear operators are open mappings or closed operators? Let's say I meet a space and an operator such as ...
1
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3answers
31 views

Prove that union of countable sets is countable

Could anyone explain to me the part in red? I can't see how the existence of the set T is used in the proof, and how theorem 2.8 is applied. Here are the relevant definitions and theorem.
10
votes
3answers
104 views

Find $\lim_{n\to\infty} \cos(\frac{\pi}{4}) \cos(\frac{\pi}{8})\ldots \cos(\frac{\pi}{2^n}) $

I already know that $$ a_n = \cos\left(\frac{\pi}{2^{n+1}}\right) = \overbrace{\frac{\sqrt{2+\sqrt{2+\ldots + \sqrt{2}}}}{2}}^{n\text{ roots}}$$ Also I know that $$\lim_{n\to\infty} 2\cos\left(\frac{...
-1
votes
2answers
28 views

Taylor series converges to $\log\frac{1+x}{1-x}$ [on hold]

How can I demonstrate that $\sum^{\infty}_{n=1}\frac{2x^{2n-1}}{2n-1}$ converges to $\log\frac{1+x}{1-x}$ for every $|x|<1$?
1
vote
0answers
21 views

McCormick envelope of two variables which are also defined in terms of an envelope

I have a equation which is defined as $\langle\langle x_ix_j\rangle^M\langle \cos(\theta)\rangle^C\rangle^M$ where $M$ is the McCormick envelope of product of variables $x_ix_j$ and $C$ is defined as ...
0
votes
1answer
30 views

Real analysis book reference

What are the best books to self study real analysis? I am a physics masters student and am looking forward to study representation theory. I want to study the real analysis I need for studying ...
2
votes
1answer
33 views

Examples of some Pointwise Convergent Sequences of Functions

I have recently come across pointwise/uniformly convergent sequences of functions, and I am hoping if someone could give some examples of certain sequences of functions so that I could understand the ...
0
votes
1answer
50 views

How to evaluate this$ \int_{0}^{c} y^{\alpha-1}(1-y)^{\beta-1}dy$?

How do I evaluate the following integral? $$ \int_{0}^{c} y^{\alpha-1} (1-y)^{\beta-1}dy $$ where $1\geq c>0$. Thank you in advance.
0
votes
0answers
28 views

Express the volume of $S^n(a)$ in terms of the volume of $B^{n-1}(a)$

I'm working on an exercise which appears in a chapter about integration on manifolds. It asks to express the volume of $S^n(a)$ in terms of the volume of $B^{n-1}(a)$. Here $S^n(a)=\{x\in\mathbb{R}^{n+...
0
votes
1answer
41 views

Why is the set $[0\le f_1\le f_2]$ measurable?

This question comes reading Analysis III of Amann and Escher. We set $$[f\le\alpha]:=\{x\in X: f(x)\le\alpha\}$$ Then it is stated that if $f_1,f_2\in\mathcal L_0(X,\lambda_n,\overline{\Bbb R}^+)$, ...
-1
votes
0answers
21 views

If $N\to \infty $ why $n(x)=\frac{1}{N}\sum_{i=1}^N \delta (x-x_i)$ can be seen as a smooth function? [on hold]

Let $\{x_1,...,x_N\}$ a collection of real number (in fact the eigen value of a matrix). If $N\to \infty $ why $n(x)=\frac{1}{N}\sum_{i=1}^N \delta (x-x_i)$ can be seen as a smooth function in $x$ ? ...
2
votes
2answers
29 views

Continuous function as pointwise limit but not as uniform limit of a sequence of continuous functions on $[0,1]$

I recentaly find an article where it is said that there is a sequence of continuous functions $\{f_n:[0,1]\rightarrow\Bbb R\}_{n\in \Bbb N}$ that converges pointqise almost everywhere to zero function ...
1
vote
1answer
28 views

Integrable function translation

For real numbers α< β and γ > 0, show that if g is integrable over [α+γ, β+γ], then $\int_{\alpha}^{\beta} g(t+\gamma)dt = \int_{\alpha+\gamma}^{\beta+\gamma} g(t)dt$ Prove this change of ...
0
votes
2answers
33 views

Is the following statement is True/false ? ..

Is the following statement is True/false ? Given that $f_n(x) =(-x)^n $ , $x \in [0,1] $ then $f_n$ converges pointwise everywhere ? i thinks it will be true same as $f(x) = 0 $ when x =0 ...
-2
votes
1answer
31 views

Proof about continuity and supremum [on hold]

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a continuous and increasing function. Prove that for all bounded non-empty set of real numbers $A$ : $\sup f[A]=f(\sup[A])$.
1
vote
6answers
120 views

Collecting proofs for $\sum_{n=2}^{\infty} \, \frac{n-1}{2^n} = 1$ [duplicate]

Update: The summation I came across has the form shown in title, and that exact question appears to be new. I could ask for proofs that take on this summation directly (without reducing it to ...
4
votes
0answers
50 views

Limits of a multiple integral function

Problem Let $f(x)\in L^{1}(\mathbb{R}^N)\cap L^{\infty}(\mathbb{R}^N)$ and $S_t=\left\{x\in\mathbb{R}^N: |x_1|\le t\right\}$ with $t>0$. Let $\phi(t)$ the integral function $$\phi(t)=\int_{\mathbb{...
-2
votes
0answers
18 views

Prove that the estimate is correct [on hold]

How to prove that the estimate is correct: $$ \sum _{n = N+1}^\infty f(n) \leq \int_{N}^\infty f(n) dn.$$ Help me, please.
1
vote
0answers
57 views

Queris related to “An arbitrary product $X=\prod_{\alpha \in J} X_\alpha$ is connected iff $X_{\alpha }$ is connected for each $\alpha\in J$.”

The following proof of An arbitrary product $X=\prod_{\alpha \in J} X_\alpha$ is connected iff $X_{\alpha }$ is connected for each $\alpha\in J$ is provided by my professor.I've some ...
4
votes
1answer
56 views

Exact value of Hausdorff measure of two dimensional Cantor set

Let $\mathcal{C}$ denote the classical Cantor set, then it is well-known that $\mathcal{C}$ has Hausdorff dimension $\alpha = \ln 2 /\ln 3$, and its $\alpha$-dimensional Hausdorff measure is $\mathcal{...
0
votes
0answers
28 views

Image of measure is relatively compact

Let $B$ be a Banach space and $\mu$ a $B$-valued measure on $X$. If $f: X \to B$ is a $\mu$-integrable function, then define $$\mu_f(E) = \int_E f d\mu$$ Is it true that the image of $\mu_f$ is ...
17
votes
1answer
231 views
+50

Show that there exist $[a,b]\subset [0,1]$, such that $\int_{a}^{b}f(x)dx$ = $\int_{a}^{b}g(x)dx$ = $\frac{1}{2}$

Let $f(x)$ and $g(x)$ be two continuous functions on $[0,1]$ and $$\int_{0}^{1}f(x) dx= \int_{0}^{1}g(x)dx = 1$$ Show that there exist $[a,b]\subset [0,1]$, such that $$\int_{a}^{b}f(x) dx= \int_{...
0
votes
0answers
30 views

Prove that the function $f(x) = \sum_{k =1}^{\infty} \frac{\sin(kx)}{2^{k}}$ is infinitely differentiable

Prove that the function $f(x) = \sum_{k =1}^{\infty} \frac{\sin(kx)}{2^{k}}$ is infinitely differentiable This is a practice problem for an exam I have coming soon. I am trying to study, but I ...
1
vote
1answer
37 views

$\int_0^{\infty}\frac{\sin x}{(1+x)^2}\,dx$ converges absolutely

This is part of problem from baby Rudin Ch 6, exer 9: How to show that $$\int_0^{\infty}\frac{|\sin x|}{(1+x)^2}\,dx$$ exists (i.e., $\lim\limits_{a\to\infty}(\int_0^{a}\frac{|\sin x|}{(1+x)^2}\,dx)...
-1
votes
2answers
56 views

Prove that $u_n =\frac{n}{n^2+1}$ is a Cauchy sequence [on hold]

Please coulde you help me in this questions: Prove that $u_n =\frac{n}{n^2+1}$ is a Cauchy sequence and $v_n=\frac{n^2}{n+1}$ is not a Cauchy sequence.
0
votes
1answer
23 views

Showing that a function has a minimum on a non compact interval

The question states: Let $f: (-1,1) \rightarrow R$ be cts and suppose $\lim_{x \to -1} f(x) = \lim_{x \to 1} f(x) = \infty$. Show that $f$ has a minimum on $(-1,1)$ My attempt: Given $\lim_{x \to -1}...
5
votes
2answers
54 views

Betweenness preserving implies monotonic?

For this question, we can assume that $f:\mathbb{R}\rightarrow\mathbb{R}$. However, I hope that an answer can generalize to arbitrary linearly ordered sets. I assume that everyone will know what I ...
0
votes
4answers
68 views

Prove the sequence $a_{1} = 4$, $a_{n + 1} = \frac{a_{n}}{2} + \frac{2}{a_{n}}$, $n = 1, 2, \ldots$ satisfies $a_{n} > 2$

Prove the sequence $a_{1} = 4$, $a_{n + 1} = \frac{a_{n}}{2} + \frac{2}{a_{n}}$, $n = 1, 2, \ldots$ satisfies $a_{n} > 2$ Let $x = a_{n}/2$. Then $a_{n + 1} = x + 1/x$. Define $f(x) = x + 1/x$ ...
0
votes
0answers
25 views

Explicit Injection of real-valued sequence space to real values

Injection requires that for some mapping $f : \mathbb{R}^\mathbb{N} \to \mathbb{R}$, $f(x) = f(y) \implies x = y$. I managed to show the existence of one by using a series of equivalences: $\mathbb{...
0
votes
1answer
27 views

Is this sum less when it contains less positive terms?

Problem Let $\{X_n\}$ be a sequence of real numbers defined by $$x_n= \left\{ \begin{array}{ccrcl} {\displaystyle\frac{1}{2k}} & \mbox{if} & {\displaystyle n} & {\...
0
votes
1answer
53 views

How to find/construct functions $f(n)$ and $g(n)$ such that $\lim_{n\to \infty} f(n)e + g(n) = 0\:?$

Here, all $f(n)$ and $g(n)$ mentioned, are functions that assume integers values for some integer $n>0$. If $n>0$ and integer, we have that $I_n = \int_0^1 x^ne^xdx = f(n)e+g(n)$ where $f(n)=(-...
0
votes
1answer
31 views

If $f$ is analytic on $(a, b)$ containing at point $x_{0}$ with $f^{(n)}(x_{0}) = 0$ for $n \in \mathbb{N}$, prove $f(x) = 0$ for all $x$.

If $f$ is analytic on $(a, b)$ containing at point $x_{0}$ with $f^{(n)}(x_{0}) = 0$ for $n \in \mathbb{N}$, prove $f(x) = 0$ for all $x$. Hi, I need help with the above problem. I'm working ...
0
votes
2answers
41 views

Can someone give me an example of a function $f$ being analytic but its power series $\sum_{n=0}^{\infty} a_{n}x^{n}$ diverging for some $x$?

is it necessarily true that $f$ being analytic implies its power series converges for all $x$? I think that it cannot diverge; however, I'm not very good at coming up with counterexamples. Can someone ...
11
votes
5answers
237 views
+150

A curiosity: how do we prove $\mathbb{R}$ is closed under addition and multiplication?

So I tried looking around for this question, but I didn't find much of anything - mostly unrelated-but-similarly-worded stuff. So either I suck at Googling or whatever but I'll get to the point. So ...
-2
votes
1answer
23 views

As I can show by definition epsilon deltra that the exponential function is continuous at point 0 [on hold]

I think I should analyze when x> 0 and when x <0 but I'm not sure a delta comes to me: ln (e + 1) and the other ln ((1/1-e))
-1
votes
1answer
73 views

Given that $\lim\limits_{n \to \infty} a_n \sum_{i=1}^n a_i^2=1$, Show $ \lim\limits_{n \to \infty} \sqrt[3]{3n}a_n=1$. [on hold]

Here is the problem: Given that $\lim\limits_{n \to \infty} a_n \sum_{i=1}^n a_i^2=1$, Show $ \lim\limits_{n \to \infty} \sqrt[3]{3n}a_n=1$.
0
votes
2answers
34 views

Injective mapping

Help me with a simple question, please If $f:X \rightarrow Y$ is a mapping, $f$ is injective if and only if for all set $Z$ and all pair of mappings $g:Z \rightarrow X$ and $h:Z \rightarrow X$, $f \...
0
votes
0answers
18 views

Frechet derivative of a function on matrix-valued $L^2$ functions

$\newcommand{\tr}{\text{Tr}}$ Let us denote $L^2(X, \mathbb C^{n \times n})$ be the space of matrix-valued $L^2$ functions. That is, if $f \in L^2(X, \mathbb C^{n \times n})$, each entry of $f$ is an $...